Descriptive Statistics > Lag Plot

## What is a Lag Plot?

A lag plot is a special type of scatter plot with the two variables (X,Y) “lagged.”

A “lag” is a fixed amount of passing time; One set of observations in a time series is plotted (lagged) against a second, later set of data. The k^{th} lag is the time period that happened “k” time points before time i. For example:

Lag_{1}(Y_{2}) = Y_{1} and Lag_{4}(Y_{9}) = Y_{5}.

The most commonly used lag is 1, called a *first-order* lag plot.

Plots with a single plotted lag are the most common. However, it is possible to create a lag plot with multiple lags with separate groups (typically different colors) representing each lag.

Lag plots allow you to check for:

**Model suitability**.**Outliers**(data points with extremely high or low values).**Randomness**(data without a pattern).**Serial correlation**(where error terms in a time series transfer from one period to another).**Seasonality**(periodic fluctuations in time series data that happens at regular periods).

## 1. Model suitability

The shape of the lag plot can provide clues about the underlying structure of your data. For example:

- A
**linear shape**to the plot suggests that an autoregressive model is probably a better choice. - An
**elliptical plot**suggests that the data comes from a single-cycle sinusoidal model.

## 2. Outliers

Outliers are easily discernible on a lag plot. The following plot shows four outliers:

## 3. Randomness

Creating a lag plot enables you to check for randomness. Random data will spread fairly evenly both horizontally and vertically. If you cannot see a pattern in the graph, your data is most probably random. On the other hand a shape or trend to the graph (like a linear pattern) indicates the data is *not *random.

The following graph shows a random pattern:

Random plots mean that there is no autocorrelation; if you know Y_{i}, you can’t begin to guess at what Y_{i-1} will be.

## 4. Serial Correlation / Autocorrelation

If your data shows a linear pattern, it suggests autocorrelation is present. A positive linear trend (i.e. going upwards from left to right) is suggestive of positive autocorrelation; a negative linear trend (going downwards from left to right) is suggestive of negative autocorrelation. The tighter the data is clustered around the diagonal, the more autocorrelation is present; perfectly autocorrelated data will cluster in a single diagonal line.

## 5. Seasonality

Data can be checked for seasonality by plotting observations for a greater number of periods (lags). Data with seasonality will repeat itself periodically in a sine or cosine-like wave.

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